Finite Group Databasebeta

← C3×#10593333 #13166753

Subgroups — C3×#10593333 #13166753

Order 1728.

372 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63822C212yesmaximal, complemented
2C2×C3×#3708C211yesmaximal, characteristic, complemented
3C3×#63736C211yesmaximal, characteristic, complemented
4C3×#63735C212yesmaximal, complemented
5C3×#63734C211yesmaximal, characteristic, complemented
6#10593333C311yesmaximal, characteristic, direct product factor
7C32×#235—22no—
8C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
9C3×#3718C2212yes—
10C3×#3715—22no—
11C3×#3713C2212yes—
12C3×#3708C2211yescharacteristic
13C3×#3708C2211yescharacteristic
14C3×#950—62no—
15#63822C2×C312yescomplemented
16C2×#3708C2×C311yescharacteristic, complemented
17#63736C2×C311yescharacteristic, complemented
18#63735C2×C312yescomplemented
19#63734C2×C311yescharacteristic, complemented
20C2×C33×C4D411yescharacteristic
21C32×#67—42no—
22C32×D12—22no—
23C32×C4×D3—42no—
24C32×Dic6—22no—
25C3×C3⋊(C3⋊C8)D411yescharacteristic
26C3×C3⋊(C3⋊C8)C2311yescharacteristic
27C3×#508—91nomaximal
28C32×#41—62no—
29C3×#235—22no—
30C3×#235—22no—
31C3×#235—42no—
32C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
33C3×#216—62no—
34C3×#215—62no—
35C3×#214—62no—
36C3×#213—62no—
37C2×C3×C3⋊C8—62no—
38#3718C22×C312yes—
39#3715—22no—
40#3713C22×C312yes—
41#3708C22×C311yescharacteristic
42#3708C22×C311yescharacteristic
43C22×C33D811yescharacteristic
44C2×C32×D3—42no—
45C33×C4D811yescharacteristic
46C33×C4C2×D411yescharacteristic
47C32×Dic3—42no—
48C3×#135—92no—
49#950—62no—
50C3×Q32—91no—
51C3×SD32—92no—
52C3×D16—91no—
53C2×C3×C16—91no—
54C32×Q8—62no—
55C32×D4—62no—
56C32×D4—122no—
57C2×C32×C4—22no—
58C2×C32×C4—42no—
59C2×C32×C4—122no—
60C2×C32×C4C3×D411yescharacteristic
61C3×#67—42no—
62C3×#67—44no—
63C3×#67—42no—
64C3×D12—22no—
65C3×D12—22no—
66C3×D12—42no—
67C3×C4×D3—42no—
68C3×C4×D3—44no—
69C3×C4×D3—42no—
70C3×Dic6—22no—
71C3×Dic6—22no—
72C3×Dic6—42no—
73C3⋊(C3⋊C8)C3×D411yescharacteristic
74C3⋊(C3⋊C8)C23×C311yescharacteristic
75C3×C3⋊C8—62no—
76C3×C3⋊C8—62no—
77#508—91noSylow
78C2×C33—21no—
79C2×C33C2×D811yescharacteristic
80C32×D3—82no—
81C3×#41—62no—
82C3×#41—62no—
83C3×#41—122no—
84#235—22no—
85C3×Q16—92no—
86C3×SD16—182no—
87C3×D8—92no—
88C2×C3×C8—91no—
89#216—62no—
90#215—62no—
91#214—62no—
92#213—62no—
93C2×C3⋊C8—62no—
94C3×C16—91no—
95C3×C16—91no—
96C22×C32—22no—
97C22×C32—42no—
98C22×C32—122no—
99C22×C32C3×D811yescharacteristic
100C2×C3×D3—42no—
101C2×C3×D3—44no—
102C2×C3×D3—42no—
103C32×C4—22no—
104C32×C4—22no—
105C32×C4—42no—
106C32×C4—42no—
107C32×C4—122no—
108C32×C4C3×D811yescharacteristic
109C32×C4C2×C3×D411yescharacteristic
110C3×Dic3—42no—
111C3×Dic3—44no—
112C3×Dic3—42no—
113#135—92no—
114Q32—91no—
115SD32—92no—
116D16—91no—
117C2×C16—91no—
118C33#50811yescharacteristic, complemented, Sylow
119C3×Q8—62no—
120C3×Q8—62no—
121C3×Q8—122no—
122C3×D4—62no—
123C3×D4—62no—
124C3×D4—122no—
125C3×D4—122no—
126C3×D4—124no—
127C3×D4—122no—
128C2×C3×C4—22no—
129C2×C3×C4—42no—
130C2×C3×C4—122no—
131C2×C3×C4—124no—
132C2×C3×C4—122no—
133C2×C3×C4GO+(4,𝔽2)11yescharacteristic
134#67—42no—
135D12—22no—
136C4×D3—42no—
137Dic6—22no—
138C3×C8—91no—
139C3×C8—91no—
140C3⋊C8—62no—
141C3⋊C8—62no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32—42no—
145C2×C32—42no—
146C2×C32—82no—
147C2×C32—242no—
148C2×C32C2×C3×D811yescharacteristic
149C3×D3—82no—
150C3×D3—84no—
151C3×D3—82no—
152#41—62no—
153Q16—92no—
154SD16—182no—
155D8—92no—
156C2×C8—91no—
157C16—91no—
158C16—91no—
159C22×C3—22no—
160C22×C3—42no—
161C22×C3—122no—
162C22×C3—124no—
163C22×C3—122no—
164C22×C3#377411yescharacteristic
165C2×D3—42no—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—42no—
169C3×C4—42no—
170C3×C4—122no—
171C3×C4—124no—
172C3×C4—122no—
173C3×C4#377411yescharacteristic
174C3×C4C2×GO+(4,𝔽2)11yescharacteristic
175Dic3—42no—
176C32—22no—
177C32—42no—
178C32C3×#50811yescharacteristic, complemented
179Q8—62no—
180D4—62no—
181D4—122no—
182C2×C4—122no—
183C2×C4C3×GO+(4,𝔽2)11yescharacteristic
184C8—91no—
185C8—91no—
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—42no—
189C2×C3—42no—
190C2×C3—82no—
191C2×C3—242no—
192C2×C3—244no—
193C2×C3—242no—
194C2×C3C2×#377411yescharacteristic
195D3—82no—
196C22—122no—
197C22C3×#377411yescharacteristic
198C4—122no—
199C4C3×#377411yescharacteristic
200C4C2×C3×GO+(4,𝔽2)11yescharacteristic
201C3—22no—
202C3—42no—
203C3#1059333311yescharacteristic, direct product factor
204C2—21no—
205C2—242no—
206C2C2×C3×#377411yescharacteristic
subgroupG/H
centerC3×C4#3774
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×D8
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#508
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#63822C2semidirect
2. C2×C3×#3708C2semidirect
3. C3×#63736C2semidirect
4. C3×#63735C2semidirect
5. C3×#63734C2semidirect
6. #10593333C3direct (N × H)
15. #63822C2×C3semidirect
16. C2×#3708C2×C3semidirect
17. #63736C2×C3semidirect
18. #63735C2×C3semidirect
19. #63734C2×C3semidirect
118. C33#508semidirect
178. C32C3×#508semidirect
203. C3#10593333direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#638222
C2×C3×#37082
C3×#637362
C3×#637352
C3×#637342
#105933333
C3×#5089